a line of 100 airline passengers is waiting to board a plane. they each hold a ticket to one of the 100 seats on that flight. (for convenience, let's say that the nth passenger in line has a ticket for the seat number n.)

unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. all of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. if it is occupied, they will then find a free seat to sit in, at random.

what is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?"

Ans: 1/2

Here is why.

At any stage there will be atmost 1 standing person (who is waiting to be seated) whose seat if already occupied.
So if at any stage, seat 1 gets occupied seat 100 will be left for the last guy, and if seat 100 gets occupied seat 1 will be left for the last guy. So basically the last guy will sit in either seat 1 or seat 100 with equal probability.